A139005 Primes p such that there exist primes p'<p"<p"'<p""<p such that the concatenation of any two among the {p,...,p""} is prime.

8389, 18433, 25253, 31231, 33647, 40289, 40357, 47237, 47303, 48731, 51721, 55621, 57331, 58763, 61129, 62303, 63601, 64189, 65657, 65677, 65983, 67723, 68491, 70099, 70571, 71341, 71741, 72739, 75653, 77153, 77641, 78509, 78511, 81401

Offset

1,1

Comments

Obviously, 2 and 5 cannot be part of such a 5-tuple.

Apart from 3, all the elements of the quintuple must have the same

residue mod 3, thus they are all = 1 or = -1 mod 6.

Two curious facts can be observed: for p=51721 and p=63601, there are

two possibilities for (p',p",p"',p""), which differ in both cases only

in p"": [p=51721, p""=44371 or 44683, p"'=22531, p"=12703, p'=8101],

resp. [63601, 26893 or 61417, 25939, 61, 7]. Secondly, the two

quintuples for p=51721 are followed by a quintuple whose sum is one of

smallest possible, only roughly half that of the two preceding

solutions.

Links

Table of n, a(n) for n=1..34.

Project Euler, Problem 60: Prime pair sets

Example

The second such 5-tuple is [18433, 12409, 2341, 1237, 7].
The two quintuplets [51721, 44371, 22531, 12703, 8101]
/*sum=139427*/ and [51721, 44683, 22531, 12703, 8101] /*sum=139739*/
are followed by [55621, 18493, 991, 883, 733] /*sum=76721*/.
The next "degenerate" case is [63601, 26893, 25939, 61, 7]
/*sum=116501*/ and [63601, 61417, 25939, 61, 7] /*sum=151025*/.
The third "degenerate" case is [71341, 63277, 54583, 7741,
241] /*sum=197183*/ and [71341, 63277, 54583, 36187, 241]
/*sum=225629*/.

Prog

(PARI) /* available from the author upon solving "Problem 60" on ProjectEuler.net */

Crossrefs

Sequence in context: A069402 A202588 A337959 * A252087 A253836 A253843

Adjacent sequences: A139002 A139003 A139004 * A139006 A139007 A139008

Keyword

nonn,base

Author

M. F. Hasler, Apr 30 2008

Status

approved